<p>Allmusic, I certainly can't prove what you see or don't see. I wonder if it would be worthwhile to take this topic to the music major forum.</p>
<p>I have no horse in this race, edad, and nothing whatsoever to prove. I just find the idea that some people don't see natural proclivity really puzzling, since it is so obvious to me.</p>
<p>I might offer it up over on the music forum....</p>
<p>The literature on gifted children suggests that they need less rather than more drill than average children in the areas of their gift.
For six years, I had a chance to observe mathematically gifted children, including a 10-year old doing A-levels (equivalent to first-year college classes here) and an 11-year old who had progressed beyond calculus and was learning quantum physics. These homeschooled kids actually spent less time on "learning" because they spent less time practicing than kids educated in school.</p>
<p>Allmusic:
I agree with you entirely about music. One of my funny memories is of S1 kicking me in rhythm with the Hallelujah chorus of the Messiah. He's the musically inclined one.</p>
<p>I think less drill, is not the same as no drill. It was one of the disadvantages of he self pace EPGY math courses. My son sometimes whizzed through things without really learning them permanently. He probably would have been allowed to skip another grade of math if he'd just remembered which was the x and which was the y axis! </p>
<p>Chess whizzes practice and study past games. I think math whizzes do the same - it's just that they aren't practicing multiplication tables any more - their practice is at a whole nuther level.</p>
<p>
[quote]
The literature on gifted children suggests that they need less rather than more drill than average children in the areas of their gift.
[/quote]
</p>
<p>Marite, I'd be really interested to know what your source is for that statement. One author I have seen quoted to that effect is Karen Rogers, and I exhaustively checked all the sources she cites for that point in her book. NONE of the sources support that conclusion--she just made it up, as far as what support she has from research. </p>
<p>If you have another source to the same effect, I will check it just as diligently. I live near a wonderful research library, and I know good research when I see it.</p>
<p>
[quote]
I think less drill, is not the same as no drill.
[/quote]
</p>
<p>The quotations about problems versus exercises are apropos here, so I'll repost them to show that many thoughtful mathematicians agree with that statement. </p>
<pre><code>"It is perhaps pertinent to make a comment or two here about the problems of the text. There is a distinction between what may be called a PROBLEM and what may be considered an EXERCISE. The latter serves to drill a student in some technique or procedure, and requires little, if any, original thought. Thus, after a student beginning algebra has encountered the quadratic formula, he should undoubtedly be given a set of exercises in the form of specific quadratic equations to be solved by the newly acquired tool. The working of these exercises will help clinch his grasp of the formula and will assure his ability to use the formula. An exercise, then, can always be done with reasonable dispatch and with a minimum of creative thinking. In contrast to an exercise, a problem, if it is a good one for its level, should require thought on the part of the student. The student must devise strategic attacks, some of which may fail, others of which may partially or completely carry him through. He may need to look up some procedure or some associated material in texts, so that he can push his plan through. Having successfully solved a problem, the student should consider it to see if he can devise a different and perhaps better solution. He should look for further deductions, generalizations, applications, and allied results. In short, he should live with the thing for a time, and examine it carefully in all lights. To be suitable, a problem must be such that the student cannot solve it immediately. One does not complain about a problem being too difficult, but rather too easy."
"It is impossible to overstate the importance of problems in mathematics. It is by means of of problems that mathematics develops and actually lifts itself by its own bootstraps. Every research article, every doctoral thesis, every new discovery in mathematics, results from an attempt to solve some problem. The posing of appropriate problems, then, appears to be a very suitable way to introduce the student to mathematical research. And it is worth noting, the more problems one plays with, the more problems one may be able to pose on one's own. The ability to propose significant problems is one requirement to be a creative mathematician."
</code></pre>
<p>Eves, Howard (1963). A Survey of Geometry volume 1. Boston: Allyn and Bacon, page ix.</p>
<pre><code>"Before going any further, let's digress a minute to discuss different levels of problems that might appear in a book about mathematics:
</code></pre>
<p>Level 1. Given an explicit object x and an explicit property P(x), prove that P(x) is true. . . .</p>
<p>Level 2. Given an explicit set X and an explicit property P(x), prove that P(x) is true for FOR ALL x [existing in] X. . . .</p>
<p>Level 3. Given an explicit set X and an explicit property P(x), prove OR DISPROVE that P(x) is true for for all x [existing in] X. . . .</p>
<p>Level 4. Given an explicit set X and an explicit property P(x), find a NECESSARY AND SUFFICIENT CONDITION Q(x) that P(x) is true. . . .</p>
<p>Level 5. Given an explicit set X, find an INTERESTING PROPERTY P(x) of its elements. Now we're in the scary domain of pure research, where students might think that total chaos reigns. This is real mathematics. Authors of textbooks rarely dare to pose level 5 problems."</p>
<p>Graham, Ronald, Knuth, Donald, and Patashnik, Oren (1994). Concrete Mathematics Second Edition. Boston: Addison-Wesley, pages 72-73.</p>
<pre><code>"First, what is a PROBLEM? We distinguish between PROBLEMS and EXERCISES. An exercise is a question that you know how to resolve immediately. Whether you get it right or not depends on how expertly you apply specific techniques, but you don't need to puzzle out what techniques to use. In contrast, a problem demands much thought and resourcefulness before the right approach is found. . . .
"A good problem is mysterious and interesting. It is mysterious, because at first you don't know how to solve it. If it is not interesting, you won't think about it much. If it is interesting, though, you will want to put a lot of time and effort into understanding it."
</code></pre>
<p>Zeitz, Paul (1999). The Art and Craft of Problem Solving. New York: Wiley, pages 3 and 4.</p>
<pre><code>". . . . As Paul Halmos said, 'Problems are the heart of mathematics,' so we should 'emphasize them more and more in the classroom, in seminars, and in the books and articles we write, to train our students to be better problem-posers and problem-solvers than we are.'
"The problems we have selected are definitely not exercises. Our definition of an exercise is that you look at it and know immediately how to complete it. It is just a question of doing the work, whereas by a problem, we mean a more intricate question for which at first one has probably no clue to how to approach it, but by perseverance and inspired effort one can transform it into a sequence of exercises."
</code></pre>
<p>Andreescu, Titu & Gelca, Razvan (2000), Mathematical Olympiad Challenges. Boston: Birkh</p>
<p>mathmom:
Indeed, I do not believe anyone advocates no drill at all. </p>
<p>Tokenadult: It is some time since I reviewed the literature on giftedness as I no longer need to advocate for my child's acceleration. When I have time, I will try to review it. But what I read was borne out by my experience with my S who needed far less practice than his peers. As I have mentioned a number of times, one of his complaints was the constant reviewing of materials already covered not only within a single year but over the years. For example the metric system was reviewed every single year through the 6th grade (and possibly beyond, but by then he had been accelerated).
When S's math teacher agreed to let him skip from 6th grade to precalc with the proviso that he would be on his own, my H carefully selected enough problems and exercizes so that he would grasp the concept and become fluent but not so many that he would be bored by repetition.
Given his love of math, which was manifested early, S has done far more math than the average student; but that is not what I would call "practice."</p>
<p>tokenadult:
I crossposted with you.</p>
<p>I agree wholeheartedly about the difference between problems and exercises. And I would repeat what I've said before: that when people talk about practicing, they mean exercises, they mean drill, they mean dribbling a basket ball 1,000 times to achieve total automaticity. That is not problem-solving.</p>
<p>Ericsson, when he talks about "deliberate practice," which is the term he prefers, is talking about something along the lines of guided problem-solving, in part, and in isolation of specific skills, in part, but it is a largely thoughtful process. Let's see if I can search my email client for an annotated quotation of him from an earlier book: </p>
<p>Ericsson, K. Anders (2003). "The Search for General Abilities and Basic Capacities: Theoretical Implications from the Modifiability and Complexity of Mechanisms Mediating Expert Performance" in Sternberg, R. & Grigorenko, E. The Psychology of Abilities, Competencies, and Expertise
pages 93-125. Cambridge: Cambridge University Press.</p>
<p>Deliberate practice is training activity "designed, typically with the help of teachers and coaches, to go just beyond the future experts' current reliable level of performance." The raised performance expectations cause the learners to make mistakes and force a continual refinement of cognitive mechanisms that mediate continued learning and
improvement (page 113). "Without deliberate practice, the performer is likely to stagnate and prematurely automate his or her performance." In various writings, Ericsson contrasts deliberate practice, a rather rare human activity, with play, that which people do for fun, and work, that
which people do for external reward. The cognitive restructuring required by deliberate practice means that it is not typically a fun activity, and the long-term payoff of increased performance from deliberate practice means that it is not usually externally rewarded in the short term. </p>
<p>That sounds a little bit like taking Math 55 to develop as a mathematician.</p>
<p>Tokenadult, I agree with you, huge difference between exercise and problem solving, but at least some times baby steps up the problem solving ladder can also double as drill. I do know that my older son learned the multiplication tables in a week and never forgot them. It probably helped that he'd figured out the principals of multiplication looking at clocks in kindergarten. My younger son was practically the last kid in his class to learn them and forgot them every summer. He needs huge amounts of drill for both arithmetic and Latin.</p>
<p>I had a talk with S about Math 55 and Math 25 (and Math 23). There were far more problem sets in Math 25 than in Math 55, but the problems were easier, he explained.
I'd interpret this as meaning that there was more "practice" in the conventional sense of the word embedded in Math 25 homework (although some familiarity with proofs was assumed whereas none was for Math 23) and there was more "doing math" in Math 55. The point of the latter was not to make the students practice writing proofs, which it was assumed they already knew, nor to solve problems. It was to get them to think about how to prove certain mathematical statements.</p>
<p>Mathmom: H refused to make S learn the multiplication tables, saying S would get them eventually without rote memorization--which he did. As you know, being good at arithmetics is no guarantee that a student will be good at math. I do think, however, that a bit more practice would have produced a little more automaticity. I have never believed in no practice; I just do not believe in practice, practice, practice. Being able to shave off 0.001 from one's speed through practice is of importance to an Olympic athlete but not to a mathematician. S always loathed timed contests and did not think they measured intellect or knowledge.</p>
<p>The line about creativity, in one of TokenAdult's posts, strikes me as interesting. There is more to math than rote computation or speed, just as there is more to music than strict note reading. Many people can be proficient at computation or note reading, but I would call neither mathematicians or real musicians. </p>
<p>Practice brings efficiency, and often profiency, but perhaps we are talking about two different things here. I think Marite knows what I mean.</p>
<p>Allmusic, I think I do. How often has it been said about some concert performance that the performer was technically proficient but had no real feeling for the music? At the risk of being flamed by his fans, this is what H says about Andrea Boccelli. </p>
<p>We recently attended some student concerts. Some of the pieces were technically very accomplished but left us unmoved. Some, however, were sung simply but produced a hush in a jam-packed concert hall. I believe that real thought about what the music meant and should convey went into that performance.</p>
<p>While there is certainly a place in the world for tenacity, hard work, practice, and self-discipline - and, those who have the above gifts often succeed where natural talent fails, it is not a substitute for talent/ability/etc.</p>
<p>Guided practice, selective problem solving, increased time spent on a beloved subject are not the same as talent, giftedness, etc. Nor is talent and giftedness necessarily the result of such. </p>
<p>Certainly, there are those who develop an expertise in mathematical reasoning with guided practice and selective problem solving. But, there are others who can quickly scan/read a math text and with no previous practice intuitively understand the concepts and solve the most difficult of problems. It is probably also a matter of learning styles.</p>
<p>Similarly, there are those born with perfect pitch, a sense of rhythm and an intuitive understanding of music. I'm not talking the mechanical reproduction of "flying fingers" on the keyboard - which is more related to muscle memory and does require sufficient practice but a fundamental understanding of music. </p>
<p>One can say the same about dance, drawing, painting, sculpting, and some sports. I've heard the same about poetry, literature, etc. </p>
<p>In the words of a coaching friend of mine, "You can't teach fast."</p>
<p>No one suggests that repetition on well established math facts will make someone a math star no matter how much time one spends practicing. As tokenadult and I have stated earlier, this is about continuous guided practice that continually evolves as the individual's repertoire changes (though some repetition on the fundamentals is often required). It is hours and hours of this continuously changing practice that produces what those of us who have not witnessed the practice often call talent. What we see is a deep understanding of math, for example, leading to working on harder and harder problems for longer periods of time with increasingly better teachers, mentors, role models, and tutors. We assume it is the "talent" that drives the process, rather than what the scientific evidence shows, it is the process that drives the "talent."</p>
<p>As for giftedness, research on early childhood language development (complexity, not type of language spoken) points to practice with vocabulary and sentence complexity from yeas 0–4 as being the best predictor of outstanding school success. This is a much better predictor than social economic status, though the two often go hand-in-hand. One can predict fairly precisely the academic success of youngsters if one simply knows the number of hours spent practicing vocabulary and language use. (See, for example, Hart & Risley, "Meaningful Differences in America's Children")</p>
<p>
[quote]
Similarly, there are those born with perfect pitch, a sense of rhythm and an intuitive understanding of music.
[/quote]
</p>
<p>I'll call blarney on this. There is no way to demonstrate any of these things at birth, and I don't believe for one second that they are as developed at birth as they are at the age when a child starts music lessons, an age by which a child has had--or not had--other experiences that tend to develop those abilities. One might as logically say, "There are those born with expressive writing ability in English." </p>
<p>I should point out that I am a father of four children, and as the husband of a piano teacher I am very aware of my children's level of "natural" talent in music, and also of their "talent" in mathematics and other subjects. They all get better as they get older, as they practice certain subjects.</p>
<p>Obviously, reflectivemom, Marite, and I see something different.</p>
<p>Truth is, when you hear an untrained vocalist, who has incredible, beautiful clarity, as well as lovely pitch and tenor, you say there IS such a thing as innate talent. All the voice lessons, (indeed, all the practice) in the world can't make a person without such a gift sing like this. </p>
<p>And school success is extremely difficult to predict, based on any measures at all. I have seen kids with IQs of 160 who end up flunking out of college, and those with much lower IQs far surpassing them. </p>
<p>This again indicates that while the gift may be there initially, it is the other qualities, such as drive, ambition, motivation, etc. (or other psychosocial issues), that typically ultimately determine the fate of talented people.</p>
<p>PS, on edit, TA. I don't have perfect pitch, but my son has had it from the moment he began piano lessons. You could play any one of 88 keys, and he could tell you exactly which note it was, even at age 6 or 7. There was no coaching or teaching, since before he was born, I didn't even know what perfect pitch was (I just assumed it was the people who didn't sing off key) ;).</p>
<p>language development, whether one's own or a foreign language, does indeed require constant practice, as I've stated earlier. I should know, as several languages I learned as a child or later in life have atrophied to nothingness as a result of lack of practice.</p>
<p>But what does guided practice means to a mathematician? or a musician? Will hours of practice turn one into Paganini (who played many wrong notes, if I recall correctly) or a technically proficient but rather boring performer? will more practice turn one into Ramamujan or Newton?</p>
<p>We seem to be going over the same grounds rather fruitlessly.</p>
<p>EDIT: cross-posted with Allmusic. Not only am I the math-challenged parent of a child who very early on, displayed a love of math but I am also the singing-off key parent of a child with perfect pitch. We did not know he had perfect pitch until much later. But we learned, after much trial and error, when he was six months that when he yelled "moo, moo" he wanted music. Specifically, Beethoven's Fifth. As for S2, we found out that math problems helped him eat his meals without fuss. Okay, so only two anecdotes. But they are the ones we lived with and had to work with.</p>
<p>"I'll call blarney on this. There is no way to demonstrate any of these things at birth, and I don't believe for one second that they are as developed at birth as they are at the age when a child starts music lessons, an age by which a child has had--or not had--other experiences that tend to develop those abilities."</p>
<p>Well, I'm certain it's going to be "dismissed" as anecdotal evidence, but my son when brought home from the hospital at less than 48 hours old would cry if not rocked exactly with the rhythm of the lullabye. Same son has perfect pitch and wrote his first musical composition - a full orchestral work - before beginning kindergarten - with each instrument written in the right key for that instrument. I have no idea how he knew to do this. We parents certainly didn't know this, pre-school yamaha group piano teacher didn't realize it either - only when showing it to other musicians did she realize what he had done. </p>
<p>Now, this same son, despite having taken art lessons in school, still draws like a 2 year old - complete with stick figures, triangle upon rectangle houses, and a circle sun with stick lines for rays. He had a friend that was drawing easily recognized faces of individuals complete with expressions before starting school. </p>
<p>I have no talent in either of these disciplines but have no problem recognizing genius in others. </p>
<p>I just wonder why some people are so quick to dismiss the talent they do not possess or see in their own children.</p>
<p>I don't agree that anyone is born with perfect pitch. It is developed through exposure to music. The untrained vocalist who performs beautifully, likewise, has been exposed to music. Formal training is not required. Music is closely linked to language development. In fact, the mother tongue music instruction method that Suzuki developed was inspired by his musings about how language proficiency was acquired. A westerner's remark about being amazed at how Japanese kids effortlessly learned the difficult Japanese language, was an epiphany for Suzuki.... Why of course they learn to speak Japanese....They're Japanese kids raised in Japan by Japanese speaking parents interracting all the time with Japanese speakers. </p>
<p>He got to thinking about how language was acquired. He then used emperical data to develop his theory & apply it to music instruction: Much one-on-one modeling from parents to their babies, positive encouragement as sounds were mastered, repeating correct pronunciation rather than criticizing mistakes. Language is acquired by ear. Reading is introduced after speaking has been mastered. No Japanese mom or dad would "give up" on their baby or toddler if mastery seemed slow, and suddenly switch to speaking German or French. Yet how many parents get exasperated by a kid's slow progress on the trombone, or with new math concepts, and just assume the kid has no musical or math aptitude? </p>
<p>Parents don't even have to think about how to "teach" language. It just happens because the babies are surrounded by language. In fact, they are immersed in it. The Suzuki music method also immerses the kids in music. I'm pretty convinced that's why my kids have perfect pitch. There was always music around them (good music) long before we decided to embrace the Suzuki method.</p>